Hmm. I see what you mean. Very neat - "What is T&E" is an on-going debate on various threads here. To me, "Logic" means the use of a recognised pattern (e.g. two matching pairs) to eliminate possibilities. So, if as Shakers suggests, there is a deeper pattern here that can be extrapolated into a rule, then that becomes part of the "Logic" armoury. In other words, as soon as you can recognise a pattern and resolve from that without having to trial it, then that is logic, not T&E.

I'm just looking at another Very Hard puzzle I'd got stuck on to see if the this idea can be applied, and therefore attempt to extrapolate a rule, as there are several pairs in various rows and columns.

I'm not having much luck... but this is the puzzle in case I'm overlooking the obvious:

This is smacking more of T&E again for me now. I thought your working had to be something along these lines.

I'd be very interested (again) for Wayne to comment on how he perceives this.

I was discussing solving these programmatically last night and began wondering whether there is a deeper element to the T&E which could be classed as logic. Looking at this puzzle I'm really beginning to want a clear definition of 'logic' that can explain these away!

Feels a bit too elastic to be a recognisable pattern - I did wonder if it was possible (based on the 7s in the first one) to describe it as a loop of an odd number of a digit exclusively linked by units with an even number of that digit. If feel this might tell you something - but I don't know what!

I spent ages last night looking for Xs and came up with a rule, but I don't think it helps. I'll put it forward anyway in case anyone else can take it further.

Consider boxes 1,2 & 3. The possibilities for the middle row in box 2 (r2c4-c6 - this is the middle of the X) must all be a subset of the possibilites for the the rest of the X - r1c1-c3, r3c1-c3, r1c7-c9 & r3c7-c9. More explicitly, it must be a subset of the horizontal and vertical pairs of corners - (r1c1-c3,r3c1-c3), (r1c1-c3,r1c7-c9), etc

Similarly, the union of possibilites for r1c4-c6 & r3c4-c6 must be the same as the union of possibilities for r2c1-c3 and r2c7-c9.

I can't find anywhere that this is useful, nor can I think of any reason why this would not be true if the better known methods of reduction have been used properly anyway. Thought I'd mention it anyway - At least it looks like an X!

Apologies. My waking/working timezones are not the same as yours in the UK, so my replies may not always appear to be prompt.

Congratulations to janders69. You have applied some wide-ranging thinking. Clearly not a man to be contained by the obvious.

First let me say that this is not the X-Wing, however. The X-Wing technique is more obvious, more obviously-logical and more self-authenticating. The X-Wing, hard though it may be, is "easier" than the technique janders69 (Jim) used.

What Jim has come across is a technique I call a "Nishio" - named for the Japanese self-styled "puzzle-master", Tetsuya Nishio. He edits a series of puzzle books in Japan. A few years ago, I found that a puzzle in one of his books was unsolveable, so I wrote to him. In that typical but admirable Japanese way, he replied with profuse and humble apologies, saying that he didn't know how it had happened, and it wouldn't happen again, etc., etc. I thought that was the end of the matter.

About a year after that, I realized that there might be a solution, after all. I went to Nishio's website and found that he was discussing the unsolveable puzzle and describing how it could be solved. I don't know if he knew all along that the puzzle could be solved and had just forgotten when he wrote to me, or whether my letter caused him to look at the puzzle again.

My Japanese is not good enough to know whether he claimed it was an acceptable logical solution or an unacceptable T&E solution. Since then, however, he has published a few puzzles which can be solved only with a "Nishio".

In the Help files for my Sudoku program, I talk of a "blurred area" between logic and T&E. I don't say so in the Help files, but I am referring to 3 or 4 techniques, one of which is the "Nishio". [To find the reference to "blurred area", go to the Index and select "Unfair puzzles/Valid but unfair puzzles".]

If a puzzle which requires a Nishio for its solution is Dubbed into my program, my program can solve it. To that extent, I recognize the puzzle as valid. However, the user is told that the puzzle is valid but arguably unfair.

However, my program will not itself create a puzzle which requires a Nishio for solution. That is because I consider that the Nishio technique is nearer to T&E than to logic.

Note that Jim used a Nishio to solve this puzzle, but he was not required to do so. He could have solved it with the easier X-Wing technique. Incidentally, it's an illustration of how difficult Sudoku puzzles are to grade.

It would be easy for my program to create puzzles that demand a Nishio for solution. The code is all there. It would be just a matter of flipping a switch in code. But my belief is that such a puzzle, though valid, is unfair.

That's because I believe a puzzle should be an entertainment, not an exercise in mechanical and tedious plotting of numbers on paper. I think it would not be possible to solve a Nishio-essential puzzle unless you have plotted all the candidates for every cell. That is, I don't think you could solve it "in your head", just by looking at the grid. (There will be the odd Super-hero who could, I guess.)

I should explain that I solve Sudokus without pencilmarks - at least, without pencilmarks entered into the cells, anyway. Perhaps I use pencilmarks, but I keep them in my head. I do it this way, partly as a challenge, and partly because I believe a pencil-and-paper, non-computer-solved puzzle shoud be capable of being solved "in the head". Otherwise, you are just emulating a computer - and very inefficiently at that.

Jim's explanation of what he did is a bit confused between a "Nishio" and a "Maunga" ... but that's a different story.

Sorry for such a long post, but if I was going to comment at all, I had to say it all.

A long, but very interesting post - thanks Wayne. Well done Jim for finding a new technique, although sadly not the elusive X-Wing!

I think the "blurred" area is an interesting problem, but from what I've read, and understand, Wayne's approach is a dashed civilised one... and as long as we keep looking at The Times we should be OK.

Having said that, it would still be interesting to know how else the puzzle we were all stuck on yesterday could have been solved without the Nishio.

I hope Wayne will forgive me if I suggest that his lengthy 'explanation' above smacks of sophistry. In effect all that talk of "Nishio" & "Maunga" merely an evasion of the central question; whether the puzzle cited by IJ can, or cannot be solved by pure logical process alone. And by logical process I mean one that involves forward sequential reasoning alone and does not allow of recursive loops or hypothetical postulates.

The reason I asked for a plainly stated solution was because I, along with other correspondents, have long ago 'solved' the puzzle - in minutes of it's being aired, by simple substitution of alternates where only two possibles present, and agree that it has a unique solution, but still dubious of the claim that it can be solved by rigorous logic. If no-one else wants such a clear resolution Wayne might be generous and mail me privately. So far he has just told me that I needn't pay much attention to the Star Wars reference - which in any case I was bound to do as having not the remotest interest or knowledge of that subject.